Sixth Mississippi State Conference on Differential Equations and Computational Simulations, Electronic Journal of Differential Equations, Conference 15 (2007), pp. 251–264.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu (login: ftp)
GAME-THEORETIC SCHEMES FOR GENERALIZED
CURVATURE FLOWS IN THE PLANE
MATTHEW RUDD
Dedicated with gratitude and best wishes to Klaus Schmitt on his 65-th birthday
Abstract. Extending some recent work by Kohn and Serfaty, we discuss a
class of two-player discrete-time games whose value functions approximate
the solutions of related geometric partial differential equations. Of particular
interest are the numerical implications of the resulting game-theoretic approximation schemes for geometric motions.
1. Introduction
N
If a hypersurface Γ0 ⊂ R moves over time with the known normal velocity V ,
then its subsequent evolution yields a family {Γt : t ≥ 0} of subsets of RN . The
subsets Γt may continue to have codimension one, or they may undergo topological
changes by breaking, merging, or fattening. Among the various techniques used to
analyze moving interfaces in the presence of such possible singularities, the level
set approach has proven very effective and leads to the geometric equations studied below. For the sake of completeness, let us recall the level set description of
propagating fronts: we look for a function u : RN × R+ → R such that
Γt = {x ∈ RN : u(x, t) = 0} ,
for t > 0 ,
by observing (cf. the derivation in [28]) that u solves the initial-value problem
ut + |Du| V = 0 ,
u(·, 0) = u0
in an appropriate sense, where the initial datum u0 : R
N
(1.1)
→ R is such that
Γ0 = {x ∈ RN : u0 (x) = 0} .
Of course, the specific nature of the level-set equation in (1.1) depends on the
form of the normal velocity function V . For the simple function V = V (x, t), for
example, we obtain the Hamilton-Jacobi problem
ut + |Du| V (x, t) = 0 ,
u(·, 0) = u0 ,
(1.2)
2000 Mathematics Subject Classification. 35K10.
Key words and phrases. Geometric level set equations; curvature flows; dynamic programming.
c
2007
Texas State University - San Marcos.
Published February 28, 2007.
251
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M. RUDD
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and more general Hamilton-Jacobi equations arise for velocity functions of the form
V = V (n, x, t), where
Du
n = n(x, t) =
(1.3)
|Du|
is the unit normal to the front at the point x at time t. There are well-known
connections between such first-order initial-value problems and deterministic games
involving either one player (i.e., optimal control problems) or two players (see, e.g.,
[2, 14, 17, 29]).
When the normal velocity V also depends on the spatial derivative of the normal
vector, i.e., the tensor
1
(I − n ⊗ n) D2 u ,
(1.4)
Dn = Dn(x, t) =
|Du|
with n given by (1.3), equation (1.1) is second-order. We are particularly interested
in normal velocities of the form
V = V (x, t, n, Dn) = − Tr [A(n, x, t)Dn]
+
+
for continuous maps A : RN ×R×R → S N , where S N
is the cone of positivedefinite, symmetric N × N matrices. Such a map A yields the initial-value problem
ut − |Du| Tr [A(n, x, t)Dn] = 0 ,
u(·, 0) = u0 .
(1.5)
The explicit expressions in (1.3) and (1.4) expose the degeneracies of this equation
more clearly than the shorthand of equation (1.5), but we write the equation in
this form for simplicity. For the special case in which A ≡ I, equation (1.5) is the
mean curvature equation,
Du (1.6)
ut − |Du| κ = 0 , where κ := div
|Du|
is the mean curvature of the front. We therefore refer to the equation in (1.5) as
the generalized curvature equation.
Beginning with the independent efforts of Evans and Spruck [18], who concentrated on the mean curvature equation, and Chen, Giga and Goto [11], who studied
equations of the form (1.7), several authors have developed a thorough viscosity
theory for geometric initial-value problems of the form
ut + F (x, t, Du, D2 u) = 0 ,
u(·, 0) = u0 ,
(1.7)
where geometric means that, for any vector p and any symmetric matrix X ∈ S N ,
F (x, t, λp, λX + µp ⊗ p) = λF (x, t, p, X)
for λ > 0 , µ ∈ R .
(1.8)
The theory for problem (1.7) also requires that F be degenerate elliptic, i.e., nonincreasing in its matrix argument:
F (x, t, p, X) ≥ F (x, t, p, Y )
when X ≤ Y .
(1.9)
Our assumptions above about the map A guarantee that the generalized curvature
equation (1.5), when written in the form (1.7), satisfies both (1.8) and (1.9). The
basic punchline is that, for a uniformly continuous initial value u0 ∈ U C(RN ),
problem (1.7) has a unique uniformly continuous viscosity solution; we refer to
[4], [6], [32] and the references therein for relevant definitions and more details.
Henceforth, we will concentrate on problems in the plane R2 .
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2. Approximating curvature flows with discrete-time games
We begin by describing the basic game studied by Kohn and Serfaty in [23];
Spencer introduced this game in [33] as a tool for combinatorial problems. To set
up the game, let the final maturity time T > 0 and objective function u0 ∈ U C(R2 )
be given. Suppose that the game begins at time t < T at the position x ∈ R2 , and
define the game’s time step ε > 0 by
T −t
ε :=
, for some integer m ∈ N .
(2.1)
m
The two players, Paul and Carol, now alternate their play as follows:
Mean curvature game.
(i) Paul chooses a unit vector v ∈ S 1 .
(i) Carol accepts or reverses√the direction v chosen by Paul.
(i) Paul moves from x to x+ 2εbv, where b = ±1 according to Carol’s previous
play.
Paul√and Carol play the next round of the game at time t + ε at the new position
x + 2ε bv. By (2.1), the game runs for precisely m rounds, after which their final
position is xT ∈ R2 and Paul pays Carol the amount u0 (xT ). Paul and Carol have
the opposing goals of minimizing and maximizing this payoff, respectively. Paul’s
value function is therefore
uε (x, t) = min{u0 (xT )},
(2.2)
where the minimum is taken over all possible final states xT that can be reached
in m plays when starting at x at time t.
Kohn and Serfaty have shown in [23] that
lim uε (x, t) = u(x, t) ,
(2.3)
ε→0
where u solves the terminal-value problem
ut + |Du| Tr [Dn] = 0 ,
u(·, T ) = u0 .
(2.4)
By reversing time, we see that u solves the initial-value problem (1.5) in the special
case in which A ≡ I, i.e., the mean curvature equation. Going backward in time,
the level sets of u therefore evolve by mean curvature and can be approximated by
the level sets of uε . We can exploit this fact to obtain a very simple algorithm for
mean curvature flow. For example, suppose that the simple closed curve Γ0 and
u0 ∈ U C(R2 ) satisfy
Γ0 = {x ∈ R2 : u0 (x) = 0} .
(2.5)
Γ0 ’s evolution by mean curvature yields the sets
Γt = {x ∈ R2 : u(x, T − t) = 0} ,
for t > 0 ,
which we can approximate with the sets
Γεt = {x ∈ R2 : uε (x, T − t) = 0} ,
ε
for t > 0 .
In particular, u (x, T − ε) will be 0 if and only if Paul can reach Γ0 in exactly
one round
midpoint of a segment of
√ of the game; this can only occur if x is the
ε
length 2 2ε whose endpoints lie on Γ0 . Similarly,
u
(x,
T − 2ε) can only be zero
√
if x is the midpoint of a segment of length 2 2ε with endpoints on Γεε , and so on.
This provides a simple, mesh-free algorithm for mean curvature flow which will be
discussed in greater detail below.
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M. RUDD
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To handle the more general geometric initial-value problem (1.5), we must modify
the basic game just described. Paul and Carol must now play as follows, where S 1
denotes the unit circle in R2 with its standard metric and v ⊥ denotes a vector
perpendicular to v relative to the standard Euclidean inner product, i.e., if v =
(v1 , v2 ), then v ⊥ := (−v2 , v1 ).
Generalized curvature game.
(i) Paul chooses a vector w ∈ S 1 , thereby defining his displacement vector
v := A(w, x, T − t)w⊥
(2.6)
as well as the pairing at (x, t):
hy, ziA := z T A(w, x, T − t)T
−1
y,
for y, z ∈ R2 .
(2.7)
(ii) Carol accepts or reverses the√
vector v defined by (2.6).
(iii) Paul moves from x to x + 2ε bv, where b = ±1 according to Carol’s
previous play.
Paul√and Carol play the next round of the game at time t + ε at the new position
x
√+ 2ε bv, and the underlying1 pairing for the next turn will come from A(w̃, x +
2εbv, T − t − ε), where w̃ ∈ S is Paul’s next choice. As before, the game runs for
precisely m rounds, after which their final position is xT ∈ R2 and Paul must pay
Carol the amount u0 (xT ). Since Paul and Carol again have the opposing goals of
minimizing and maximizing this payoff, respectively, Paul’s value function is still
uε (x, t) = min{u0 (xT )},
(2.8)
where the minimum is taken over all possible final states xT that can be reached
in m plays when starting at x at time t.
As with the mean curvature game analyzed in [23], we want to understand the
behavior of the value functions uε , as ε → 0, for this generalized game. To analyze
their limit, we rely on the fact that each value function uε satisfies the dynamic
programming equation
n
o
√
(2.9)
uε (x, t) = min1 max uε (x + 2ε b1 v1 , t + ε) ,
w1 ∈S b1 =±1
where the vectors v1 and w1 are related by (2.6); the subscripts here emphasize the
fact that
√ b1 , w1 and v1 correspond to Paul’s first play. A similar equation holds for
uε (x + 2ε b1 v1 , t + ε), thereby advancing from time t + ε to time t + 2ε :
n
o
√
√
uε (x, t) = min1 max min1 max uε (x + 2ε b1 v1 + 2εb2 v2 , t + 2ε) .
w1 ∈S b1 =±1 w2 ∈S b2 =±1
Continuing in this manner m times yields
uε (x, t) = min1 max · · · min 1 max
w1 ∈S b1 =±1
wm ∈S bm =±1
m
n
o
√ X
uε (x + 2ε
bi vi , T ) .
(2.10)
i=1
After m rounds, however, we reach the end of the game, where uε (·, T ) equals the
known terminal value u0 . We may therefore rewrite (2.10) in the form
uε (x, t) = min1 max · · · min 1 max
w1 ∈S b1 =±1
wm ∈S bm =±1
m
n
o
√ X
u0 (x + 2ε
bi vi ) ,
i=1
(2.11)
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obtaining a representation formula for uε as a function of u0 . In terms of the
operator
S(ε) : U C(R2 ) → U C(R2 )
defined by
√
(S(ε)u) (x) := min1 max u(x + 2ε bv) ,
(2.12)
w∈S b=±1
formula (2.11) becomes
m
uε = (S(ε)) u0 ,
and we want to determine
m
m
lim (S(ε)) u0 = lim (S(ε)) u0 .
m→∞
(2.13)
ε→0
We will see that this limit is the unique solution u of the terminal-value problem
ut + |Du| Tr [ A(n, x, T − t)Dn ] = 0 , u(·, T ) = u0 ,
(2.14)
N +
N +
where A : R × R × R → S
is continuous and S
denotes the cone of
positive-definite, symmetric N × N matrices. Before proving this fact, let us briefly
describe its numerical implications. Defining the set Γ0 by (2.5) as before, the sets
Γt correspond to the generalized curvature flow of Γ0 , and the sets Γεt are their
approximations via the value function uε . The point x will belong to Γεε if and only
if Paul can reach Γ0 in one play, i.e., if and only if there exists w ∈ S 1 such that
√
x ± 2εA(w, x, T − ε)w⊥ ∈ Γ0 .
N
This leads to a conceptually simple algorithm for generalized curvature flow, but
the nonlinear dependence on w certainly complicates its implementation. We will
come back to numerical issues below in Section 3.
The following theorem analyzes the limit (2.13).
Theorem 2.1. Let u0 ∈ U C(R2 ) be given, let u be the unique viscosity solution of
−t
(2.14), and define ε := Tm
for m ∈ N and t < T . Then
m
m
u(·, t) = lim (S(ε)) u0 = lim (S(ε)) u0 .
m→∞
(2.15)
ε→0
Proof. According to the general convergence results of Barles and Souganidis ([5];
see also [30],[31]), modified as in [3] to handle the degeneracy in equation (2.14),
we must show that our approximation scheme is stable, monotone and consistent.
In particular, we must verify that S(·) has the following four properties:
(i) S(0) = I,
(ii) S(ε) (u + c) = S(ε)u + c, for any constant c,
(iii) S(ε)u ≤ S(ε)ũ whenever u ≤ ũ, and
(iv) for any smooth φ and any x ∈ R2 ,
(S(ε)φ) (x) − φ(x) lim
= |Dφ(x)| Tr [A(n, x, T − t)Dn]
(2.16)
ε→0
ε
if Dφ(x) 6= 0, where
n=
Dφ(x)
|Dφ(x)|
and Dn =
1
(I − n ⊗ n) D2 φ(x) ,
|Dφ(x)|
(2.17)
while
λ ≤ lim inf
ε→0
(S(ε)φ) (x) − φ(x) ε
≤ lim sup
ε→0
(S(ε)φ) (x) − φ(x) ε
≤Λ
(2.18)
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M. RUDD
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if Dφ(x) = 0, where λ and Λ are the extreme eigenvalues of D2 φ(x).
The first two of these properties are trivial. The third follows easily from the
definition of S(ε): for u, ũ ∈ U C(R2 ) with u ≤ ũ, let w ∈ S 1 be optimal for ũ with
v = A(w, x, T − t)w⊥ . Then
√
√
(S(ε)u) (x) ≤ max u(x + 2εbv) ≤ max ũ(x + 2εbv) = (S(ε)ũ) (x) ,
b=±1
b=±1
verifying the monotonicity of the scheme.
To check the scheme’s consistency (property (iv)), pick any smooth φ and a point
x ∈ R2 , and suppose, first of all, that Dφ(x) 6= 0. Recall the Riemannian structure
on R2 defined by (2.7) and Taylor expand φ to obtain
o
n
√
(S(ε)φ) (x) = min1 max φ(x + 2εbv)
w∈S b=±1
n
o
√
= min1 max φ(x) + 2εbhDφ(x), viA + εhD2 φ(x)v, viA + o(ε3/2 ) ,
w∈S b=±1
from which see that
n √2b
o
(S(ε)φ) (x) − φ(x)
= min1 max √ hDφ(x), viA + hD2 φ(x)v, viA + o(ε1/2 ) .
ε
w∈S b=±1
ε
Choosing
Dφ(x)
,
(2.19)
w=
|Dφ(x)|
the corresponding displacement vector v is
1
⊥
v=
A(w, x, T − t) (Dφ(x))
(2.20)
|Dφ(x)|
and hDφ(x), viA = 0; any choice of w for which the latter does not hold will clearly
not be optimal for small ε. By the definition of the underlying pairing (2.7), we
have
T
hD2 φ(x)v, viA = v T A(w, x, T − t)−1 D2 φ(x)v ,
and a direct calculation using (2.19) and (2.20) reveals that
hD2 φ(x)v, viA = |Dφ(x)| Tr [A(n, x, T − t)Dn] ,
with n and Dn given by (2.17). Property (iv) follows directly in this case. If,
instead, Dφ(x) = 0, consistency is even simpler, as the first-order term in the Taylor
expansion vanishes and the remaining second-order term satisfies (2.18).
The Taylor expansion used in this proof is essentially the heuristic presented
in [23], but the machinery developed by Barles and Souganidis [5] renders this
argument perfectly rigorous. In addition to being short and simple, the resulting
proof seems more natural from the point of view of standard viscosity techniques
than the proof offered in [23].
3. Algorithms and examples
There are by now numerous papers devoted to algorithms for curvature-dependent flows. Before discussing our game-theoretic algorithm in detail, we will briefly
summarize some of the extant approaches, postponing a thorough comparison of
these methods for a future publication.
The first schemes for mean curvature and related motions relied on finite difference methods on uniform grids ([1],[9],[13]), and subsequent work focused on
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257
developing faster, more efficient algorithms. In [24], Bence, Merriman and Osher
introduced a very successful algorithm, diffusion-generated motion by mean curvature, which Evans [15] and Barles and Georgelin [3] subsequently analyzed. Ishii,
Pires and Souganidis [22] then consolidated the work of Bence, Merriman and Osher with that of Gravner and Griffeath on models of cellular automaton dynamics
[20], resulting in the more general framework of threshold dynamics type approximation schemes. Ruuth and Merriman provide an interesting discussion of some
of these developments in [27]; they point out, for example, that diffusion-generated
motion by mean curvature amounts to tracking the center of a circle as it traverses
the curve in such a way that precisely half of its area is always inside the curve.
Described this way, we see that the Bence-Merriman-Osher scheme has a similar
geometric flavor to our game-theoretic scheme. It seems much simpler, however, to
track the midpoint of a segment than to track the center of a circle while maintaining this area condition. A further simplifying feature of our scheme is that it does
not require a grid for computations.
Motivated by developments in mathematical morphology and image processing,
Catté et al. [10] developed a different approach which leads to the same min-max
operator used here. Their point of view is quite different from ours and from that
in [23], however, and, unlike ours, their implementation is grid-based. Following
the earlier work by Evans [15], Catté et al. prove their analogue of Theorem
2.1 with nonlinear semigroup techniques; in semigroup parlance, we may interpret
equation (2.15) as an exponential formula of Crandall-Liggett type ([12],[26]), with
the scheme S(ε) playing the role of the resolvent operator. In addition to their
differing philosophy and implementation, we note that Catté et al. do not consider
the general geometric equation (2.14) to which the game-theoretic approach applies.
There are at least two other noteworthy algorithms for curvature-dependent
flows which have appeared recently. Oberman [25] has proposed and analyzed
a grid-based scheme for mean curvature motion which, in the language of image
processing, is a simple and efficient implementation of a median filter. Cao and
Moisan [8], on the other hand, have developed an interesting algorithm which relies
on the geometry of the curve and does not involve a computational grid; we refer to
Cao’s book [7] for more on this algorithm and the role of curve evolution in image
processing.
3.1. Mean curvature flow. Having briefly surveyed these other techniques, we
now describe the algorithms inspired by the two-player games above, beginning with
the simpler case of mean curvature motion. To approximate the mean curvature
flow of a planar curve Γ0 , we proceed as follows, retaining the notation Γt and Γεt
used earlier.
Idealized mean curvature algorithm.
(i) Choose the time step ε > 0.
(ii) Having computed the approximation Γεjε for some positive integer j, find
all pairs (x1 , x2 ) ∈ Γεjε such that
√
(3.1)
dist (x1 , x2 ) = 2 2ε ,
and, for each pair, add the point
x1 + x2
x̄ :=
2
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M. RUDD
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to the new set Γε(j+1)ε .
(iii) Stop after k rounds when there is no pair (x1 , x2 ) ∈ Γεkε satisfying (3.1).
This is an idealized scheme for several reasons. First, we must somehow discretize
the original curve Γ0 in order to perform computations, and one cannot expect
pairs of points from the resulting discrete set to satisfy (3.1) exactly. Second, if
Γ0 ’s evolution by mean curvature results in fattening, it is not clear how to find
all pairs satisfying (3.1) efficiently. Since a specific example of the latter situation
may be useful, suppose for the moment that Γ0 is the left-hand set in Figure 1. It
Figure 1. The set on the left fattens as it evolves by its mean
curvature, yielding sets with interior like the one on the right. The
set on the right is solid, like a steering wheel with four holes.
is well-known that the set on the left will fatten as it evolves by its mean curvature
(it also appears, for instance, in [16]); one way to see this, in fact, is to apply the
idealized algorithm above by hand on a piece of scrap paper. The set on the right
is typical of the resulting evolution. From the point of view of the mean curvature
game, fattening occurs when the trajectory of the traversing segment’s midpoint is
not unique. Accommodating this possibility complicates the implementation of the
algorithm, so we keep things simple in the present paper by assuming that Γ0 is a
simple closed curve. Future work will address a more robust implementation.
Instead of discretizing with a grid of some sort, we will approximate the curves
Γεjε directly with discrete sets, initializing computations with a discrete set ∆ε0 of
points belonging to the initial curve Γ0 . This leads to the following straightforward
implementation of the scheme described above.
Practical mean curvature algorithm.
(i) Choose the time step ε > 0.
(ii) Suppose that, for some integer j ≥ 0, we have a discrete set ∆εjε which
approximates the curve Γεjε . We assume that the points in ∆εjε are sorted,
corresponding to an oriented traversal of Γεjε .
(iii) Begin at the first point x0 ∈ ∆εjε . Let x1 and x2 be the first consecutive
pair of points in ∆εjε such that
√
√
dist (x1 , x0 ) < 2 2ε and dist (x2 , x0 ) > 2 2ε ,
(3.2)
and define
x̄ :=
2x0 + x1 + x2
.
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259
The point x̄ is the midpoint of the segment whose endpoints are the midpoints of the segments connecting x0 and x1 and x0 and x2 , respectively.
Add x̄ to the discrete approximation ∆ε(j+1)ε of the curve Γε(j+1)ε .
(iv) Advance to the second point in ∆εjε and repeat the preceding procedure,
adding a second point to ∆ε(j+1)ε . Continue in this way until all of ∆εjε has
been exhausted, thereby completing ∆ε(j+1)ε . (When dealing with points
at the end of ∆εjε , the points x1 and x2 will come from the beginning of
∆εjε . This can be handled easily with a simple array data structure.)
(v) Stop after k rounds when (3.2) cannot be satisfied by points in ∆εkε .
Figures 2 and 3 illustrate the application of this mean curvature algorithm to
two sets, a unit circle and a nonconvex curve built out of four half-circles of radius
1. Both computations used a segment of length 0.1, corresponding to the time step
ε = 1/800, and each snapshot indicates the corresponding number of rounds of the
game. The initial discretization ∆ε0 of the circle consisted of 48 uniformly spaced
points, and it took 0.48 seconds to compute all 338 rounds of its resulting evolution.
For the nonconvex curve, ∆ε0 consisted of 96 points, its evolution to extinction lasted
for 1163 rounds, and it took 5.79 seconds to compute its full evolution.
It is well-known that a simple closed curve will convexify, become circular, and
shrink to a point as it evolves by mean curvature ([19],[21]), a fact borne out by
these examples. (This provides a quick but crude check that the algorithm works
as it should.) Although we have avoided the possibility of fattening, it is worth
emphasizing that this algorithm does not require any convexity.
3.2. Generalized curvature flow. To approximate the generalized curvature flow
of Γ0 according to
√ equation (2.14), we cannot simply track the midpoint of a segment of length 2 2ε as it traverses the curves Γεjε . Instead, we must modify the
mean curvature algorithm as follows.
Idealized generalized curvature algorithm.
(i) Choose the time step ε > 0.
(ii) Having computed the approximation Γεjε for some positive integer j, find
all pairs (x1 , x2 ) ∈ Γεjε such that the problem
√
2ε A(w, x̄, T − (j + 1)ε)w = v
(3.3)
has a solution w ∈ S 1 , where
v := x2 − x1
and x̄ :=
x1 + x2
.
2
(3.4)
For each such pair (x1 , x2 ), add the point x̄ to the new set Γε(j+1)ε .
(iii) Stop after k rounds when (3.3) has no solution w ∈ S 1 for any pair (x1 , x2 ) ∈
Γεkε .
This is an idealized scheme for the same reasons given above in the case of mean
curvature flow. A practical implementation in this case is much more challenging,
however, since we must deal with the nonlinear problem (3.3). At this preliminary
stage, we only present results for the trivial generalized curvature flow problem in
which
A(n, x, t) ≡ A
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M. RUDD
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Evolution of circle by curvature
1
1
1
0.5
0.5
0.5
0
0
0
−0.5
−0.5
−0.5
−1
−1
0
Initial circle
1
−1
−1
0
1
...after 40 rounds...
−1
−1
1
1
1
0.5
0.5
0.5
0
0
0
−0.5
−0.5
−0.5
−1
−1
0
1
...after 120 rounds...
−1
−1
0
1
...after 160 rounds...
−1
−1
1
1
1
0.5
0.5
0.5
0
0
0
−0.5
−0.5
−0.5
−1
−1
0
1
...after 240 rounds...
−1
−1
0
1
...after 280 rounds...
−1
−1
0
1
...after 80 rounds...
0
1
...after 200 rounds...
0
1
...after 320 rounds...
Figure 2. Snapshots of the motion of a circle by its mean curvature.
for some constant nonsingular matrix A. In this very special case, we simply change
step (ii) of the practical mean curvature algorithm as follows: starting with the first
point x0 ∈ ∆εjε , find the first consecutive points x1 and x2 in ∆εjε such that
√
√
kA−1 (x1 − x0 ) k < 2 2ε and kA−1 (x2 − x0 ) k > 2 2ε ,
(3.5)
where k · k is the standard Euclidean norm. Add the point
x̄ :=
2x0 + x1 + x2
4
to ∆ε(j+1)ε and, continuing in this manner, complete the set ∆ε(j+1)ε .
Figures 4 and 5 illustrate the application of this generalized curvature algorithm
to the circle and nonconvex curve treated earlier, again using a segment of length
0.1 in both examples. The matrix A used in both cases is
1 0
A=
,
0 4
EJDE-2007/CONF/15
GAME-THEORETIC SCHEMES FOR CURVATURE FLOWS
261
Evolution of nonconvex curve by curvature
2
2
2
1
1
1
0
0
0
−1
−1
−1
−2
−2
0
Initial curve
2
−2
−2
0
2
...after 25 rounds...
−2
−2
2
2
2
1
1
1
0
0
0
−1
−1
−1
−2
−2
0
2
...after 175 rounds...
−2
−2
0
2
...after 300 rounds...
−2
−2
2
2
2
1
1
1
0
0
0
−1
−1
−1
−2
−2
0
2
...after 800 rounds...
−2
−2
0
2
...after 1050 rounds...
−2
−2
0
2
...after 75 rounds...
0
2
...after 550 rounds...
0
2
...after 1160 rounds...
Figure 3. Snapshots of the motion of a nonconvex curve by its
mean curvature.
and one sees clearly that both curves approach “circles” relative to the underlying
pairing (2.7) defined by A.
Future work will confront the more difficult problem (3.3), as well as numerical
analysis questions which we have ignored here (e.g., convergence rates and the
interaction of the time step ε with the resolution of the initial discretization ∆ε0 ).
The application of these ideas to other problems, such as the motion of curves in
R3 and the active contour model in [9], will also be discussed elsewhere.
Acknowledgments. I discussed the main ideas presented here in an invited talk
at the 6th UAB-MSU Conference on Differential Equations and Computational
Simulations, held at Mississippi State University in May 2005. It is a pleasure to
thank the conference organizers for all of their hospitality and hard work, and it was
a tremendous honor to participate in the special session honoring Klaus Schmitt, a
wonderful advisor and friend. Happy birthday, Klaus!
262
M. RUDD
EJDE/CONF/15
Evolution of circle by generalized curvature
1
1
1
0.5
0.5
0.5
0
0
0
−0.5
−0.5
−0.5
−1
−1
0
Initial circle
1
−1
−1
0
1
...after 15 rounds...
−1
−1
1
1
1
0.5
0.5
0.5
0
0
0
−0.5
−0.5
−0.5
−1
−1
0
1
...after 45 rounds...
−1
−1
0
1
...after 60 rounds...
−1
−1
1
1
1
0.5
0.5
0.5
0
0
0
−0.5
−0.5
−0.5
−1
−1
0
1
...after 90 rounds...
−1
−1
0
1
...after 105 rounds...
−1
−1
0
1
...after 30 rounds...
0
1
...after 75 rounds...
0
1
...after final round.
Figure 4. Snapshots of the motion of a circle by generalized curvature.
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Matthew Rudd
Department of Mathematics, University of Texas at Austin, Austin, TX 78712, USA
E-mail address: rudd@math.utexas.edu